Here is an critical tip for before your next recording session -- and it cannot be over-stated -- is take the time to deaden your drummer's kit before you record. Schedule a few hours -- at least as much as you would spend on one rehearsal, and do NOTHING but get those drums tuned and quiet. Often, the drummer will want to change heads before recording. That's fine. That's appropriate. But unless your drummer is a total, anal, analytical madman about drum tuning and drum sound, don't let him tune his own drums alone in a vacuum. One or two other bandmates should be there helping/guiding him and acting as extra ears, and thinking about how the drums will record.
Think about it this way -- close mics -- and I don't mean just sensitive condensers -- even dumb 57s and 58s, pick upeverything. Both the stuff you want, and the stuff you don't want. If you're in a big studio with a ton of mics, and you're micing each drum, you'll minimally put TWO on the snare -- one on top and one below. If you're using the Glyn Johns approach like we do, that's at least 6 mics, and still, TWO of them are on the snare. If you use 4 mics, you'll likely have one on the snare, one on the kick, and two overheads. And if you're just using two overheads only, that snare ping is STILL gonna cut through and hurt.
When the drummer buys new heads for a session, make sure he also buys some drum dampeners. The gels, the clips, the felt things, the clear plastic rings -- get a variety of stuff to try, and have some gaffer tape on hand as well. Taping the heads (or taping things like paper towel to the heads) still works in a pinch.
It's important for all the drums, but it's most important thing is the snare drum, because it is hit so frequently and it sets the beat for every song. If its pinging like a bitch, it will drive you mad later.
You have to tune both the top and bottom snare heads. We used the Bonham approach -- you tune it to F# on the bottom head, D# on the top -- resulting overall tone of G# (when the snares aren't engaged). But there are other good approaches as well, and loads of videos on YT on how to do them, so there is no excuse not to do it. Then you've got to get the snare itself seated correctly, so that it rattles as little as possible when you hit adjacent drums. Once that's done, and you engaged the snares, if he's hitting that snare and you still hear audible ping with a note value, you should deaden it. Once the snares are engaged, that snare should be 99% "crack" with very little note value. Gotta fix any flubby or tubby kick drum sounds as well, but that's comparatively easier than the snare because it's a simpler drum by design, and you may only use (or mic) the beater head.
It is tedious. Your drummer may likely fight you to a degree and say "that's too dead." Don't relent. If you don't fix it before-hand, you'll be stuck trying to EQ it out of every damn track. I've done it on a whole album, and it was both time-consuming, less than 100% effective -- you never completely kill it, and carving out the offending frequencies can be detrimental to the overall drum sound. And FWIW, my drummer's snare was a pingy snare by design. He recently bought a new metal snare that sounds MUCH better, and was easier to tame. And our new tracks reflect the improvement. But we still had to go through the tuning/deadening process. The good news is that once done, when you hear the drum tracks unmixed, they already sound great before there's any processing on them, and we will need far less time mixing them.